Mechanics of Advanced Materials and Structures

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Mechanics of Advanced Materials and Structures

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Analytical Approach to Free Vibrations of Cracked Timoshenko Beams Made of Functionally Graded Materials

Hamid Zabihi Ferezqia; Masoud Tahania; Hamid Ekhteraei Toussia a Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran Online publication date: 18 June 2010

To cite this Article Ferezqi, Hamid Zabihi , Tahani, Masoud and Toussi, Hamid Ekhteraei(2010) 'Analytical Approach to

Free Vibrations of Cracked Timoshenko Beams Made of Functionally Graded Materials', Mechanics of Advanced Materials and Structures, 17: 5, 353 — 365 To link to this Article: DOI: 10.1080/15376494.2010.488608 URL: http://dx.doi.org/10.1080/15376494.2010.488608

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Mechanics of Advanced Materials and Structures, 17:353–365, 2010 Copyright © Taylor & Francis Group, LLC ISSN: 1537-6494 print / 1537-6532 online DOI: 10.1080/15376494.2010.488608

Analytical Approach to Free Vibrations of Cracked Timoshenko Beams Made of Functionally Graded Materials Hamid Zabihi Ferezqi, Masoud Tahani, and Hamid Ekhteraei Toussi

Downloaded By: [B-on Consortium - 2007] At: 13:13 21 June 2010

Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

This paper presents an analytical investigation of the free vibrations of a cracked Timoshenko beam made up of functionally graded materials (FGMs). It is assumed that the beam is constructed of FGM materials with a power law variation of metal-ceramic volume fraction. The perspective of wave method is adopted for the analysis. The method considers the nature of the propagation and reflection of the waves along the beam. Consequently, the propagation, transmission and reflection matrices for various discontinuities located on the beam are derived. Such discontinuities may include crack, boundaries or change in section. By combining these matrices a global frequency matrix is formed. In order to investigate the effect of the beam’s structural synthesis, different natural frequencies are obtained and studied. Keywords

Wave method, free vibrations, Timoshenko beam, FGM, crack

1.

INTRODUCTION Cracked structures have always attracted significant interest by researchers. Cracks in a structural element can reduce the natural frequencies and change the vibration mode shapes due to local flexibility introduced by the crack. The dynamics of cracked structural members, especially beams, has been the subject of many research works due to growing interest in non-destructive damage evaluation of engineering structures using modal responses (natural frequencies and mode shapes) of a structure in the past two decades. Numerous attempts to quantify local defects are reported in the literature. In general, there exist three basic crack models, namely the equivalent reduce section model, the local flexibility model from fracture mechanics and the continuous crack flexibility model [1]. One of the most common approaches is the finite element method where the key point is to define the stiffness matrix as the inverse of the compliance matrix [2]. To establish the local flexibility of the beam due to the presence of a crack the method of finite element and fracture mechanics are employed.

Received 1 June 2009; accepted 3 October 2009. Address correspondence to Masoud Tahani, Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran. E-mail: [email protected]

FGMs are the novel class of macroscopic composites with spatially continuous materials which have attracted considerable research efforts over the past few years due to their increasing applications in many engineering sectors [3]. The technique of grading ceramics along with metals initiated by the Japanese material scientist in Sendai has marked the beginning of exploring the possibility of using FGMs for various structural applications [3]. Numerous studies have been conducted on FGM beams, plate and shell structures, dealing with a variety of subjects such as thermal stresses [4], fracture analysis [5, 6], static bending [7], free vibration and dynamic responses [8], buckling and postbuckling [9], piezo-thermo-elastic behavior [9] and so on. A literature review shows that there are quite a few papers presenting crack and fracture analyses of FGM structures. Another aspect of this study is the vibration analysis. Various approaches have been applied in vibration analysis of cracked beams. Such approaches include finite element approach [10], Galerkin and local Ritz approach [11], approximate analytical approach [12], transfer matrix approach [13] and dynamic stiffness matrix approach [14]. Among the specific works done in this area one can point to the paper by J. Yang et al. [15] which provides an analytical study of free and forced vibration of inhomogeneous Euler-Bernoulli beams containing open edge cracks. In their analysis the beam is assumed to be subjected to an axial compressive force and a concentrated transverse load moving along the longitudinal direction. Over the years, many analytical techniques have been developed for treating wave propagation problems. Central among these is the method of Fourier synthesis or the spectral analysis [16], where the behavior of the signal is viewed as a superposition of many infinitely long wave trains of different periods. Moreover Sridhar et al. [17] studied the wave propagation analysis in anisotropic and inhomogeneous structures using pseudospectral finite element method. Graff [18] studied the free vibrations of a cracked Timoshenko beam from the wave standpoint in which the vibrations are described in terms of wave propagation, transmission and reflection in waveguides. The reflection and transmission characteristics of flexural vibration waves have been studied by a number of researchers (e.g., see [19]) and the reflection and transmission matrices for various discontinuities on a cracked

353


354

H. Z. FEREZQI ET AL.

Timoshenko beam have also been derived by Mei et al. [20]. Ke et al. [21] studied the free vibration and elastic buckling of FGM beams containing open edge cracks based on Timoshenko beam theory. In this paper, the free vibration of a FGM cracked Timoshenko beam is investigated by using the wave method. The beam is constructed by using FGM materials with a power law variation of metal-ceramic volume fraction. The application of power law variation of material properties is mainly dependent on its usual usage in industrial applications. The crack is modeled by a rotational spring. Also, the matrices of wave method are derived for the various types of probable discontinuities which may locate on the beam by using the equations of motion for the FGM Timoshenko beam. Such discontinuities include cracks, boundaries and change in section. These matrices can be combined to provide a concise and systematic approach to free vibration analysis of FGM cracked Timoshenko beams. Based on the devised model, typical parametric analyses are performed and the influence of different discontinuities on the beam upon the natural frequencies is studied. The results are provided to examine the effects of crack depth, crack location, total number of cracks, slenderness ratio, change of section and change of power law index of the material property distribution on the vibration characteristics of the cracked FGM Timoshenko beam. 2.

MATHEMATICAL FORMULATIONS It is intended to determine the natural frequencies of a cracked FGM Timoshenko beam with dimensions l, b and h as the length, width and height of the beam, respectively, and a as the depth of an edge crack, as depicted in Figure 1. It is assumed that the material is isotropic, and the grading is assumed to be only through the thickness. 2.1. Displacement Field and Strains Here the FGM beam will be studied within the framework of the first-order shear deformation beam theory. The displacement field of the beam may be represented as u(x, y, z, t) = u(x, t) + zψ(x, t), w(x, y, z, t) = w(x, t).

v(x, y, z, t) = 0, (1)

In Eq. (1) the rotation function ψ (about the y axis) and the transverse displacement w are functions of the longitudinal

coordinate x and time t and z denotes the thickness coordinate. Also u(x, t) represents the axial displacement component of the middle surface of the beam. It is to be noted that for transverse vibration of the beam u(x, t) can be neglected throughout the analysis. The linear strain-displacement relations of elasticity are given by [22] ∂ψ , εy = εz = 0, ∂x ∂w . = ψ+ ∂x

εx = z γxz

γxy = γyz = 0, (2)

2.2. Constitutive Relations Consider a functionally graded beam, which is made from a mixture of ceramics and metals. It is assumed that the composition properties of FGM vary through the thickness of the beam. The variation of material properties can be expressed as p(z) = (pt − pb ) Vt + pb ,

(3)

where p denotes a generic material property like modulus and pt and pb denote the corresponding properties of the top and bottom faces of the beam, respectively. Also Vt in Eq. (3) denotes the volume fraction of the top face constituent and follows a simple power-law as Vt = (z/ h + 1/2)n where h is the total thickness of the beam, z is the thickness coordinate (−h/2 ≤ z ≤ h/2), and n (0 ≤ n ≤ ∞) is a parameter that dictates the material variation profile through the thickness. Here we assume that moduli E and G and density ρ vary according to Eq. (3) and the Poisson’s ratio ν is assumed to be a constant. The linear constitutive relations are ⎫ ⎡ ⎧ ⎤⎧ ⎫ Q11 Q12 0 ⎪ ⎬ ⎨ σx ⎪ ⎨ εx ⎬ ⎥ ⎢ σy = ⎣ Q12 Q22 0 ⎦ εy , ⎪ ⎭ ⎪ ⎩ ⎭ ⎩ γxy 0 0 Q66 σxy Q44 0 γyz σyz = , (4) 0 Q55 σxz γxz where Q11 = Q22 =

E(z) , 1 − ν2

Q44 = Q55 = Q66 =

Q12 =

νE(z) , 1 − ν2

E(z) = G(z). 2(1 + ν)

(5)

2.3. Equations of Motion Using Hamilton’s principle the following equations of motion are obtained FIG. 1. Geometry of FGM beam and possible variation of ceramic and metal through the thickness.

∂Qx ¨ = I1 w, ∂x

∂Mx ¨ − Qx = I3 ψ, ∂x

(6)

355

FREE VIBRATIONS OF CRACKED TIMOSHENKO BEAMS

where the force and moment resultants and the mass terms are defined as Qx = (I1 , I3 ) =

h/2

−h/2 h/2

σxz dz,

Mx =

h/2 −h/2

σx zdz,

ρ(1, z2 )dz.

(7)


∂w , Qx = A55 ψ + ∂x

Mx = D11

∂ψ , ∂x

(8)

h/2

−h/2 h/2

−h/2

h/2

−h/2 h/2

Q55 dz = Q11 dz =

where a1+ , a2+ , a1− and a2− denote the wave amplitudes of w and a¯ 1+ , a¯ 2+ , a¯ 1− and a¯ 2− denote the wave amplitudes of ψ. From Eq. (12), W and are related to each other by the following relation: A55 K 2 − I1 ω2 I1 ω2 . (15) =i = iK 1 − W A55 K A55 K 2 Hence, the relation between the wave amplitudes a and a¯ are as follows:

where

D11 =

w(x) = a1+ e−iK1 x + a2+ e−K2 x + a1− eiK1 x + a2− eK2 x , ψ(x) = a¯ 1+ e−iK1 x + a¯ 2+ e−K2 x + a¯ 1− eiK1 x + a¯ 2− eK2 x , (14)

−h/2

In Eqs. (6) a dot over displacement components indicates partial differentiation with respect to temporal variable t. The boundary conditions consist of specifying w or Qx and ψ or Mx at x = 0 and x = L. Upon substitution of Eqs. (4) into Eqs. (7), the force and moment resultants in terms of displacement components will be obtained which can be presented as follows:

A55 =

By solving this equation a set of wave numbers are derived that are functions of the frequency ω as well as the properties of structure and are in the form of ±K1 and ±K2 . Waves in the beam travel in both positive and negative directions as the ± sign before K1 and K2 indicates. Now with the time dependence eiωt suppressed, the solution to Eq. (10) can be written as

−h/2

E(z) dz, 2(1 + ν)

a¯ 1+ = iP , a1+

E(z) 2 z dz. 1 − ν2

(9)

a¯ 2+ = N, a2+

a¯ 2− = −N. (16) a2−

where I1 ω2 , P = K1 1 − A55 K 2

Lastly, the governing equations of motion are obtained by substituting Eqs. (8) into Eqs. (6) ∂ψ ∂ 2 w = I1 w, + A55 ¨ ∂x ∂x 2 ∂w ∂ 2ψ ¨ = I3 ψ. D11 2 − A55 ψ + ∂x ∂x

a¯ 1− = −iP , a1−

I1 ω2 N = K2 1 + . A55 K 2

(17)

3.

(10)

2.4. Free Vibration Analysis In order to obtain the natural frequencies of the beam we consider a harmonic solution as w(x, t) = W e−iKx e−iωt ,

ψ(x, t) = e−iKx e−iωt , (11)

√ where i = −1, ω is the natural frequency and K is the wave number. Substitution of Eqs. (11) into Eqs. (10) and rewriting the equations in the matrix form yields

A55 K − I1 ω iK A55 2 K + A55 − I3 ω −iK A55 2

D11

THE PROPAGATION, REFLECTION AND TRANSMISSION OF WAVES From a wave standpoint, vibrations propagating along a beam component are reflected and transmitted upon discontinuities and boundaries. The propagation is governed by the so-called propagation matrix. Consider two points A and B on a flexurally vibrating uniform beam at distance x apart. Denoting the positive and negative-going wave vectors at points A and B as a+ and a− , and b+ and b− , respectively, they are related by

2

2

W

+

a =

= 0.

(13)

(18)

where

(12) For non-trivial solution the determinant of Eq. (12) should be set to zero which gives us the following second-order polynomial in K 2 as a0 K 4 + b0 K 2 + c0 = 0.

b+ = f(x) a+ , a− = f(x) b− ,

−

b =

a1+ a2+ b1− b2−

,

−

a =

a1− a2−

,

+

b =

b1+ b2+

, (19)

.

and f(x) =

e−iK1 x

0

0

e−iK2 x

,

(20)

356


Three common boundary conditions of interest are simply supported, clamped and free boundaries. Corresponding to these boundaries, KT and KR are either zero or infinite. The reflection matrices for simply supported, clamped and free boundary conditions are presented as follows: rs =

−1 0 , 0 −1

1 rf =

rf11 rf12 rf21 rf22

⎤ P − iN −2iN ⎢ P + iN P + iN ⎥ ⎥, rc = ⎢ ⎣ −2P P − iN ⎦ − P + iN P + iN ⎡

(26)

,

FIG. 2. A general end side boundary of the beam.


where which is known as the propagation matrix for a distance x. The reflection and transmission characteristics are governed by the reflection and transmission matrices. The following derives the reflection and transmission matrices at discontinuities such as boundaries and cracks. 3.1. Reflection at Boundaries A general boundary is shown in Figure 2. The incident waves a+ give rise to reflected waves a− which are related by −

+

a = ra .

(21)

The reflection matrix r can be determined by considering equilibrium at the boundary that is Qx = KT W,

Mx = −KR ψ,

(22)

where KT and KR are the translational and rotational stiffnesses of the support, respectively. If the boundary is at x = 0, then the equilibrium conditions become +

−

α11 a = α22 a ,

(23)

where α11 = α22 =

D11 PK1 + iKR P A55 iP − A55 iK1 − KT

−D11 NK2 + KR N A55 N − A55 K2 − KT

−D11 PK1 + iKR P A55 iP − A55 iK1 + KT

D11 NK2 + KR N A55 N − A55 K2 + KT

rf21 = 2A55 D11 PK1 i (P − K1 ) , rf22 = −A55 D11 [PK1 (N − K2 ) − NK2 i (P − K1 )] , = −A55 D11 [PK1 (N − K2 ) + NK2 i (P − K1 )] . (27)

3.2. Crack In this study, the local flexibility model from finite element method and fracture mechanics are adapted. Cracks could be open or breathing (open and close in time), depending on the loading conditions and vibration amplitudes. The open crack model is valid throughout this paper. To this aim after calculating the value of stress intensity factor by using ANSYS software, the strain energy release rate can be derived by J =G=

K2 , Etip

(28)

where Etip is the modulus of elasticity of the FGM beam at the crack tip and K and J are the stress intensity factor and the strain energy release rate, respectively. Also we have

θM(a) 2

1 dM =− θ , 2 da

(29)

, .

Hence, the reflection matrix is r=

rf12 = −2A55 D11 NK2 (N − K2 ) ,

∂U d G=− =− ∂a da

(24)

α−1 22 α11 .

rf11 = A55 D11 [PK1 (N − K2 ) + NK2 i (P − K1 )] ,

where U, M, θ and a are the strain energy, bending moment, slope due to bending and crack depth respectively. The slope due to bending is related to bending moment by θ = CMwhere C is the compliance of the beam at the crack position. Therefore, by the following equation the compliance can be derived:

(25)

a 0

Gda = −

θM θ2 =− . 2 2C

(30)

357


That is, β21 b+ + β22 a− = β23 a+ ,

(37)

where

FIG. 3.

β21

A transverse open edge crack.

Now considering an open crack at x = 0 as shown in Figure 3, a set of positive-going waves a+ is incident upon the crack and gives rise to transmitted and reflected waves b+ and a− which are related to incident waves through the transmission and reflection matrices t and r by


b+ = t a+ ,

a− = r a+ .

(31)

The transverse displacements and the slopes of the beam on the left and right hand sides of the crack are

β22

D11 K1 P −D11 K2 N = β23 = , iA55 (P − K1 ) A55 (N − K2 ) −D11 K1 P D11 K2 N = , iA55 (P − K1 ) A55 (N − K2 )

(38)

Equations. (31), (34) and (37) can be solved to obtain the transmission and reflection matrices at the crack discontinuity as −1 β23 − β22 β−1 t = β21 − β22 β−1 12 β11 12 β13 , −1 β23 − β21 β−1 r = β22 − β21 β−1 11 β12 11 β13 .

(39)

wL = a1+ e−iK1 x + a2+ e−K2 x + a1− eiK1 x + a2− eK2 x ,

wR = b1+ e−iK1 x + b2+ e−K2 x , −iK1 x iK1 x ψL = iPa+ + N a2+ e−K2 x − iPa− − N a2− eK2 x , 1e 1e

−iK1 x −K2 x ψR = iPb+ + Nb+ . 1e 2e

(32)

Since the beam is continuous we have wR = wL ,

ψR = ψL + CD11

∂ ψL , ∂x

(33)

where the term CD11 ∂∂ψxL represents a jump in the bending slope caused by local flexibility change at the crack. C is related to crack ratio µ, which is defined as the ratio between the depth of the crack and the thickness of the beam (i.e., µ = a/ h). The continuity condition can be written in matrix form as follows: β11 b+ + β12 a− = β13 a+ ,

(34)

where β11 = β12 = β13 =

1 1 iP N

3.3. Change in Section Consider two beams of different properties are joined at x = 0 as shown in Figure 4. Due to impedance mismatching, incident waves from one beam give rise to reflected and transmitted waves at the junction. The reflection and transmission matrices can then be obtained from the continuity and equilibrium conditions. Denoting the parameters related to the incident and transmitted sides of the junction with subscripts L and R, respectively, choosing the origin at the point where the sections changes, at x = 0 we have w L = wR ,

ψL = ψR ,

QxL = QxR . (40)

Equations (31) and (40) can be put into matrix form in terms of the reflection and transmission matrices rLL and tLR as γ21 tLR + γ22 rLL = γ23 ,

M L = MR ,

γ12 tLR + γ11 rLL = γ13 , (41)

,

−1 iP − K1 PCD11 1 iP + K1 PCD11

−1 , N + K2 NCD11 1 . N − K2 NCD11

(35)

Furthermore, the equilibrium of the crack gives us: QxL = QxR ,

ML = MR ,

(36)

FIG. 4.

A stepwise change in section.

358


where

-5

γ21 = γ22 = γ23 =

The equations can be solved for the reflection and transmission matrices rLL and tLR , which are given as −1 γ13 − γ11 γ−1 tLR = γ12 − γ11 γ−1 22 γ21 22 γ23 , −1 γ13 − γ12 γ−1 rLL = γ11 − γ12 γ−1 21 γ22 21 γ23 .


x 10

n= n= n= n=

4.5 4

0.05 0.5 1 2

3.5

Compliance (C)

γ11 =

1 1 −1 −1 −1 −1 , γ12 = , γ13 = , iPL NL iPR NR iPL NL (D11 )R K1R PR −(D11 )R K2R NR , i(A55 )R (PR − K1R ) (A55 )R (NR − K2R ) −(D11 )L K1L PL (D11 )L K2L NL , i(A55 )L (PL − K1L ) (A55 )L (NL − K2L ) (D11 )L K1L PL −(D11 )L K2L NL . (42) i(A55 )L (PL − K1L ) (A55 )L (NL − K2L )

5

3 2.5 2 1.5 1 0.5 0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Crack depth ratio (a / h)

(43)

The reflection and transmission matrices for the waves run into various types of discontinuities can be combined to provide a concise and systematic approach for the vibration analysis of FGM Timoshenko beams with various discontinues. 4.

NUMERICAL RESULTS The solution procedure outlined in the previous section is applied to the free vibration of three functionally graded beams. One is a cantilevered cracked beam and the second one is a cantilevered cracked stepped beam and the next one is a multiple edge cracked cantilever FGM beam. It is assumed that the lower surface of the beam is rich of ceramic and the top surface is rich of metal. The mechanical properties of the metal and ceramic are given in Table 1 [23]. The length-to-thickness ratio or slenderness ration (i.e., L/ h) is assumed to be 25 in all numerical examples. Also the total thickness of the plate (h) is considered to be 0.04 m.

FIG. 5. (a/ h).

Change of compliance corresponding to various crack depth ratio

with different material constant n. Clearly, as the a/ h ratio is increased the numerical value of the compliance is also increased accordingly. 4.2. Example 1. A Uniform Beam with a Crack Figure 6 shows a uniform cracked beam. The geometric discontinuity is located at point B. The incident and reflected waves at the clamped boundary A, the left and right hand sides of the crack discontinuity, B and the free end side, C are denoted by − + − + + a+ , a− , b+ 1 , b1 , b2 , b2 , c and c , respectively. The relationships between the incident and the reflected waves at the boundaries are described as a+ = rA a+ ,

c− = rC c+ ,

where rA and rC are the reflection matrices in points A and C, respectively.

4.1.

Change of Compliance Due to the Presence of a Crack In this section, the effect of the crack depth ratio (a/ h) upon the compliance is studied. Figure 5 shows the change of compliance as a result of various crack depth ratios in FGM beams TABLE 1 The mechanical properties of the metal and ceramic components [23] Metal: Ti-6Al-4v

Ceramic: ZrO2

E = 66.2 GPa ν = 0.321 ρ = 4.41 × 103 kg/m3

E = 117 GPa ν = 0.321 ρ = 5.6 × 103 kg/m3

(44)

FIG. 6.

A cracked cantilever beam.

359


TABLE 2 Natural frequencies (rad/s) of a uniform beam with various crack depth ratio Crack ratio µ = 0.1

Crack ratio µ = 0.3

Mode number

Present

FEM

Present

FEM

Present

FEM

1 2 3

162.8 826.4 3148.3

163.1 828.4 3153.2

93.8 568.2 3147.5

93.9 570 3153.9

54.4 527.9 3147.4

54.9 529.4 3153.1

At the geometric discontinuity B, these relationships are as follows: − + b+ 2 = r b2 + t b1 ,


Crack ratio µ = 0.5

+ − b− 1 = r b1 + t b2 ,

(45)

where r and t are the reflection and transmission matrices of the crack. The propagation relations are, a− = f(L1 )b− 1, − = f(L ) c , b− 2 2

+ b+ 1 = f(L1 ) a , + c = f(L2 ) b+ 2,

(46)

where f(L1 ) and f(L2 ) are the propagation matrices between AB and BC sections, respectively. Equations (44)–(46) can be rewritten in matrix form as ⎡

−I

⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢ ⎢ f(L1 ) ⎢ ⎢0 ⎢ ⎢ ⎣0 0

rA 0 0

0 0 r

0 0 −I

0 0

t 0 −I 0

0 0 0

0 0 t

0 rC 0

0 −I 0

−I 0

t 0

0 0

0 0

−I 0 0 0

f(L1 ) 0 0 0 f(L2 ) 0

0

0

0

0

0 0 −I 0

−I 0

⎤⎡

a+

⎤

⎥ ⎢ a− ⎥ ⎥⎢ ⎥ ⎥⎢ +⎥ ⎥ ⎢ b1 ⎥ ⎥⎢ ⎥ ⎥ ⎢ b− ⎥ ⎥⎢ 1 ⎥ ⎥⎢ +⎥ ⎥ ⎢ b2 ⎥ ⎥⎢ ⎥ ⎥ ⎢ b− ⎥ ⎥⎢ 2 ⎥ ⎥⎢ + ⎥ ⎦⎣c ⎦

f(L2 )

= 0.

c− (47)

For a non-trivial solution the determinant of the coefficient matrix must vanish. From this condition the natural frequencies of the beam can be found. The crack is assumed to be at x = L/2.

The values of the natural frequencies are obtained through a computer program written in MATLAB. The first three natural frequencies of the FGM beam with the power law index 2 (i.e., n = 2) are listed in Table 2 and compared with the results obtained by using ANSYS software. Figures 7a, 7b and 7c show the first three natural frequencies of the cracked FGM beam down to various crack locations with crack ratio µ = 0.3 and various values of n. Also the first mode shape of the cracked FGM beam with n = 2 and the crack location at x = L/2 for different values of µ are shown in Figure 7d. To verify the validity of the solution developed in this paper, in a special case the material is assumed to be homogeneous and the solution is compared with that given by Mei et al. [20]. The physical properties of the material are chosen as those in [20] which are listed in Table 3. It is assumed that the crack is located in three different locations, x = 0.3L, x = 0.5L and x = 0.8L with the same crack ratio µ = 0.3. The present results are tabulated in Table 4. It is seen that there is excellent agreement between the present results and those of Mei [20]. Figure 8 shows the effects of crack depth ratio a/ h and slenderness ratio L/ h on the first natural frequency of the beam. The results indicate that natural frequency is decreased as the crack depth ratio and slenderness ratio are increased. Figure 9 shows the effects of crack depth ratio on the first three natural frequencies of the sample beam once the crack is located at x = 0.5L. The results reveal that in general natural frequency is decreased as the depth of crack is increased. However, if the crack is located on the node of a vibrating mode shape of an un-cracked beam, it is found from the obtained results that, the crack has no effect on the respected natural frequency of the beam.

TABLE 3 Physical properties of a sample homogeneous beam [20] Rigidities Shear: GAk (N) 6343.3 Mass per unit length (kg/m) 0.0544

Bending: EI (Nm2 )

Torsion: GJ (Nm2 )

0.2865 Width: b (m) 0.0127

0.1891 Depth: h (m) 0.00318

Polar mass moment of inertia (kgm) 0.777e-6 Length: L(m) 0.1905

Poisson’s ratio 0.29

360


TABLE 4 Natural frequencies of a sample homogeneous cracked beam Crack located at x = 0.3L Mode number 1 2

Crack located at x = 0.5L

Present

Mei et al. [20]

Present

Mei et al. [20]

Present

Mei et al. [20]

34.2 215.4

34.1 215.2

35.1 214.6

35.2 214.3

35.4 215.3

35.3 215.4

− + c+ 2 = rLL c2 + tRL c1 ,

1200

180

1100

Second natural frequency (ωc)

First natural frequency (ωc)

and transmission at the step change, which are related as the following:

200

160 140 120 100 80 n= n= n= n=

60 40

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.05 0.5 1 2 0.9

+ − c− 1 = rRR c1 + tLR c2 .

n= n= n= n=

1000

(48)

0.05 0.5 1 2

900 800 700 600 500 400

1

0

0.1

0.2

Crack location ( L 1 / L )

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


(a)

(b)

3200

1

3000

0.9 0.8

2800

0.7

2600

Mode shape

Third natural frequency (ωc)


4.3. Example 2. A stepped Beam with a Crack Figure 10 shows a cracked stepped beam with step discontinuity at point C. The analysis follows the same procedure as described above, except that there is additional wave reflection

20

Crack located at x = 0.8L

2400 2200

0.6 0.5 0.4

2000 0.3 n= n= n= n=

1800 1600 1400

0

0.1

0.2

0.3

0.4

0.5

0.05 0.5 1 2 0.6

0.1 0.7


(c) FIG. 7.

a/h = a/h = a/h = a/h =

0.2

0.8

0.9

1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.2 0.4 0.5

0.9

1

Non dimensional length of beam

(d)

The natural frequencies of a cracked FGM beam and its first mode shape. (a) first, (b) second and (c) third natural frequencies. (d) first mode shape.

361


The subscripts of r and t identify incident and transmitted sides of the junction. The propagation relations are redefined as:

3000 L / h = 6.25 L / h = 12.5 L / h = 25


2500

a− = f(L1 ) b− 1, − b− = f(L ) c , 2 2

2000

1500

− c− 2 = f(L3 ) a ,

c+ = f(L3 ) a+ 2,

(49)

1000

The relationship between the incident and the reflected waves at the end boundary is described as

500

0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

d− = rD d+ ,

0.5


FIG. 8. The effects of the crack depth ratio and slenderness ratio on the first natural frequency.

1200

160

0.05 0.5 1 2

n= n= n= n=

1100


n= n= n= n=

180

140 120 100 80

1000

60

0.05 0.5 1 2

900 800 700 600 500

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

400

0.5

0

0.05

0.1

0.15


0.2

n= n= n= n=

3200 3150

0.05 0.5 1 2

3100 3050 3000 2950 2900 2850

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


FIG. 9.

0.25

0.3

0.35


3250


40

(50)

where rD is the reflection matrix in point D.

200



+ b+ 1 = f(L1 ) a , + + c = f(L2 ) b2 ,

The effect of crack depth ratio on the first three natural frequency.

0.4

0.45

0.5

362


For a non-trivial solution the determinant of the coefficient matrix must vanish. From this condition the natural frequencies of the beam can be found. In this example, a crack located at x = L/3 and step change is assumed at x = 2L/3. Figure 11 shows the first two natural frequencies of the beam corresponding to various crack depth and various values of n. It is seen that both frequencies are decreased with increasing the crack depth ratio. 4.4. FIG. 10.

Example 3. A Multiple Edge Cracked Cantilever Beam In this example the vibration characteristics of a cantilever FGM beam with six cracks are studied. The analysis follows the same procedure as described in the first Example. In this case the cracks are located at points A, B, C, D, E, F and G and the phenomena of wave reflection and transmission are repeated at

A cracked stepped cantilever beam.


Writing Eqs. (44), (48), (49) and (50) in matrix form gives: ⎡

−I

rA

0

0

0

0

0

0

0

0

0

0

⎤⎡

a+

⎤

⎢ ⎥⎢ ⎥ 0 0 0 0 0 0 0 0 0 rD −I ⎥ ⎢ a− ⎥ ⎢ 0 ⎢ ⎥⎢ +⎥ ⎢ ⎥ ⎢ 0 0 r −I 0 t 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ b1 ⎥ ⎢ ⎥⎢ −⎥ 0 t 0 −I r 0 0 0 0 0 0 ⎥ ⎢ b1 ⎥ ⎢ 0 ⎢ ⎥⎢ +⎥ ⎢ ⎥ ⎢ 0 0 0 0 0 0 rLL −I 0 tRL 0 0 ⎥ ⎢ ⎥ ⎢ b2 ⎥ ⎢ ⎥⎢ −⎥ 0 0 0 0 0 tLR 0 −I rRR 0 0 ⎥ ⎢ b2 ⎥ ⎢ 0 ⎢ ⎥ ⎢ + ⎥ = 0. ⎢ f(L1 ) 0 −I ⎥⎢c ⎥ 0 0 0 0 0 0 0 0 0 ⎢ ⎥⎢ 1 ⎥ ⎢ ⎥⎢ ⎥ −I 0 f(L1 ) 0 0 0 0 0 0 0 0 ⎥ ⎢ c− ⎢ 0 ⎥ ⎢ ⎥ ⎢ 1+ ⎥ ⎢ ⎢ 0 ⎥ 0 0 0 f(L2 ) 0 −I 0 0 0 0 0 ⎥ ⎢ c2 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ 0 0 0 0 −I 0 f(L2 ) 0 0 0 0 ⎥ ⎢ c− ⎢ 0 2 ⎥ ⎢ ⎥⎢ +⎥ ⎢ ⎥ ⎢ 0 0 0 0 0 0 0 0 f(L3 ) 0 −I 0 ⎥ ⎣ ⎦⎣d ⎦ 0 0 0 0 0 0 0 0 0 −I 0 f(L3 ) d−

240

960


200

0.05 0.5 1 2

n= n= n= n=

940


n= n= n= n=

220

180 160 140 120 100 80 60 40

(51)

920

0.05 0.5 1 2

900 880 860 840 820 800 780

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35


(a)

0.4

0.45

0.5

760

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35


(b)

FIG. 11. The first two natural frequencies of the cracked beam. (a) first and (b) second natural frequencies.

0.4

0.45

0.5

363


every point. Writing Eq. (18) for the propagation of waves, as well as Eq. (21) for the reflection at the boundaries and Eq. (31) for the reflection and transmission of waves at each crack in matrix form gives:


⎡

For a non-trivial solution, the determinant of the coefficient matrix must vanish. In this example, cracks are located at x = L/7, x = 2L/7, x = 3L/7, x = 4L/7, x = 5L/7 and x = 6L/7 simultaneously.

−I rc 0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 I 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

I f 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 t 0 −I r

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 r −I 0

t

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0

f

0 −I 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0

0 −I f

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0

0

0

t

0 −I r

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0

0

0

r −I 0

t

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0

0

0

0

0

f

0 −I 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0

0

0

0

0

0 −I f

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0

0

0

0

0

0

0

t

0 −I r

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0

0

0

0

0

0

0

r −I 0

t

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0

0

0

0

0

0

0

0

0

f

0 −I 0

0

0

0

0

0

0

0

0

0

0

0

0 0 0

0

0

0

0

0

0

0

0

0 −I f

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0

0

0

0

0

0

0

0

0

0

0

t

0 −I r

0

0

0

0

0

0

0

0

0

0 0 0

0

0

0

0

0

0

0

0

0

0

r −I 0

t

0

0

0

0

0

0

0

0

0

0 0 0

0

0

0

0

0

0

0

0

0

0

0

0

f

0 −I 0

0

0

0

0

0

0

0

0 0 0

0

0

0

0

0

0

0

0

0

0

0

0

0 −I f

0

0

0

0

0

0

0

0

0 0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

t

0 −I r

0

0

0

0

0

0 0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

r −I 0

t

0

0

0

0

0

0 0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

f

0 −I 0

0

0

0

0 0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 −I f

0

0

0

0

0 0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

t

0 −I r

0

0 0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

r −I 0

t

0 0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

f

0 −I

0 0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 −I f

0 0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

⎢ ⎢f ⎢ ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎣ 0

0

0 rf

⎤⎧ ⎫ a+ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎪ − ⎪ ⎪ ⎪ 0 ⎥⎪ a ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎥⎪ + 0 ⎥⎪ ⎪ b1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎥ − ⎪ ⎪ 0 ⎥⎪ ⎪ b ⎪ ⎪ 1 ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎥ + 0 ⎥⎪ b2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ b− ⎪ ⎪ 0 ⎥ ⎪ ⎪ ⎪ ⎥⎪ 2 ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ + ⎪ ⎪ 0 ⎥ ⎪ c ⎪ 1 ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎥⎪ − ⎪ ⎪ ⎪ 0 ⎥⎪ c ⎪ ⎪ 1 ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ + 0 ⎥⎪ c ⎪ 2 ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ c− ⎪ ⎪ 0 ⎥ ⎪ ⎥⎪ ⎪ ⎪ 2 ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎥ +⎪ ⎪ 0 ⎥⎪ d ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ d+ ⎪ ⎪ 0 ⎥ ⎪ ⎪ ⎥⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ − ⎪ ⎪ ⎥ 0 ⎥⎪ ⎪ d ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎬ 0 ⎥ ⎨ d− ⎪ 2 ⎥ =0 ⎥ 0 ⎥⎪ e1+ ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎥ ⎪ e1+ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎥ − ⎪ 0 ⎥⎪ e ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎥ − ⎪ 0 ⎥⎪ ⎪ e ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ + ⎪ 0 ⎥ ⎪ f1 ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎥⎪ +⎪ ⎪ ⎪ 0 ⎥⎪ f1 ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ − 0 ⎥⎪ f ⎪ 2 ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪f− ⎪ ⎪ 0 ⎥ ⎪ ⎥⎪ ⎪ ⎪ 2 ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎥ +⎪ ⎪ 0 ⎥⎪ g ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ +⎪ 0 ⎥ ⎪ ⎪ g ⎥⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ −⎪ ⎪ ⎪ 0 ⎥ ⎪ g ⎪ 2 ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎥⎪ ⎪ g− ⎪ ⎪ 0 ⎥⎪ ⎪ ⎪ 2 ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ +⎪ 0 ⎥⎪ ⎪ h ⎪ ⎪ ⎪ ⎦⎪ ⎪ ⎪ ⎩ −⎭ −I h (52) 0

364

H. Z. FEREZQI ET AL. 200

1200 n= n= n= n=

180

n= n= n= n=

1000 Second natural frequency (ωc)


160

0.05 0.5 1 2

140 120 100 80 60

0.05 0.5 1 2

800

600

400

40 200 20 0

0

0.05

0.1

0.15

0.2 0.25 0.3 0.35 Crack depth ratio (a / h)

0.4

0.45

0.5

0

0

0.05

0.1

0.15


(a)

0.4

0.45

0.5

(b)


3500 n= n= n= n=


3000

0.05 0.5 1 2

2500

2000

1500

1000

500

0

0

0.05

0.1

0.15


0.4

0.45

0.5

(c) FIG. 12.

The first three natural frequencies of the cracked beam. (a) first, (b) second and (c) third natural frequencies.

Figure 12 shows the first three natural frequencies of the beam down to various crack depths and various values of n. It is seen that by increasing the crack depth ratio, all the first three frequencies are decreased. 5.

CONCLUSIONS An analytical approach based on the wave method is presented to study the free vibrations of a FGM Timoshenko beam with various types of structural discontinuities. To this ends, within the framework of the first-order shear deformation beam theory and by using Hamilton’s principle, equations of motion are obtained. Henceforward, the matrices of the wave method are derived for various discontinuities such as the cracks, boundaries and the change in cross section. The effect of crack presence is modeled by a rotational spring and an equivalent stiffness factor is estimated and used accordingly. It is shown through

numerical examples that by introducing the propagation, reflection and transmission matrices, the process of vibration analysis become systematic and concise. The results confirm that the natural frequency of a cracked beam is reduced by increasing its crack depth ratio. Also it is shown that the location of crack and the power low index have significant effects on the natural frequencies of the beam. Especially it is shown that the more the slenderness factor of a beam the less its natural frequency. REFERENCES 1. A.D. Dimarogonas, Vibration of Cracked Structure, Eng. Fracture Mech., vol. 55(5), pp. 831–857, 1996. 2. D.Y. Zheng and N.J. Kessissooglou, Free Vibration Analysis of a Cracked Beam by Finite Element Method, J. Sound Vib., vol. 273, pp. 457–475, 2004. 3. M. Koizumi, The Concept of FGM Ceramic Transactions, Functionally Graded Materials, vol. 34(1), pp. 3–10, 1993.


FREE VIBRATIONS OF CRACKED TIMOSHENKO BEAMS 4. N. Noda, Thermal Stresses in Functionally Graded Materials, J. Thermal Stresses, vol. 22(4–5), pp. 477–512, 1999. 5. Z.H. Jin and R.C. Batra, Some Basic Fracture Mechanics Concepts in Functionally Graded Materials, J. Mech. Phy. Solids, vol. 44(8), pp. 21–35, 1996. 6. F. Erdogan and B.H. Wu, The Surface Crack Problem for a Plate with Functionally Graded Properties. J. Appl. Mech. ASME, vol. 64(3), pp. 448–506, 1997. 7. J. Yang and S.H. Shen, Nonlinear Bending Analysis of Shear Deformable Functionally Graded Plates Subjected to Thermo-Mechanical Loads Under Various Boundary Conditions, Compos Part B, vol. 34(2), pp. 103–150, 2003. 8. J. Yang and S.H. Shen, Dynamics Response of Initially Stressed Functionally Graded Thin Plates, Compos Struct., vol. 54(4), pp. 497–508, 2001. 9. K.M. Liw, J. Yang, and S. Kitipornchai, Postbuckling of Piezoelectric FGM Plates Subject to Thermo-Electro-Mechanical Loading, Int. J. Solid Struct., vol. 40(38), pp. 69–92, 2003. 10. M. Krawczuk and W.M. Ostachowicz, Modeling and Vibration Analysis of a Cantilever Composite Beam with Transverse Open Crack, J. Sound Vib., vol. 183(1), pp. 69–89, 1995. 11. M.H.H. Shen and C. Pierre, Free Vibration of Beam with Single-edge Crack, J. Sound Vib., vol. 170(2), pp. 237–259, 1994. 12. Y. Narkis, Identification of Crack Location in Vibrating Simply Supported Beam, J. Sound Vib., vol. 172(4), pp. 549–558, 1994. 13. T.C. Tsai and Y.Z. Wang, Vibration Analysis and Diagnosis of Cracked Shaft, J. Sound Vib., vol. 192(3), pp. 607–620, 1996.

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14. N.T. Khiem and T.V. Lien, The Dynamic Stiffness Matrix Method in Forced Vibration Analyses of Multiple-Cracked Beam, J. Sound Vib., vol. 254(3), pp. 541–555, 2002. 15. J. Yang, Y. Chen, Y. Xiang, and X.L. Jia, Free and forced vibration of cracked inhomogeneous beams under an axial force and a moving load, J. Sound Vib., vol. 312, pp. 166–181, 2008. 16. J.F. Doyle, Wave Propagation in Structures, second edition, Springer-Verlag Series, Berlin, 1989. 17. R. Sridhar, A. Chakraborty, and Gopalakrishnan, Wave propagation analysis in anisotropic and inhomogeneous structures using pseudospectral finite element method, Int. J. Solid Struct., vol. 43, pp. 4997–5031, 2006. 18. K.F. Graff, Wave Motion in Elastic Solid, Ohio State University Press, Ohio, 1975. 19. B.R. Mace, Wave Reflection and Transmission in Beams, J. Sound Vib., vol. 97, pp. 237–246, 1984. 20. C. Mei, Y. Karpenko, S. Moody, and D. Allen, Analytical Approach to Free and Forced Vibrations of Axially Loaded Cracked Timoshenko Beams, J. Sound Vib., vol. 291, pp. 1041–1060, 2006. 21. L.-L. Ke, J. Yang, S. Kitipornchai, and Y. Xiang, Flexural Vibration and Elastic Buckling of Cracked Timoshenko Beams Made of Functionally Graded Materials, Mech. Adv. Mater. Struct., vol. 16, pp. 488–502, 2009. 22. Y.C. Fung, Foundations of Solid Mechanics, 1st ed., Englewood Cliffs, New Jersey: Prentice–Hall, 1965. 23. J.N. Reddy and C.D. Chin, Thermomechanical Analysis of Functionally Graded Cylinder and Plates, J. Therm. Stresses, vol. 21, pp. 593–626, 1998.